In my experience it is worth considering variants of Lawvere fixed point theorem. In the present case, I would split things up as follows, in order to circumvent the non-constructive nature of Brouwer's fixed point theorem.
Also, let me point out that we need not worry about exponentials too much, even though they do not exist in the category of topological spaces, unless the exponent is nice enough. We can move over to a cartesian-closed subcategory, such as teh compactly generated spaces, or to a cartesian closed supcategory, such as equilogical spaces.
Theorem: [Approximate Lawvere] Suppose $(B, d)$ is a metric space and $e : A \to B^A$ is a continuous map, such that for every continuous map $g : A \to B$ and $\epsilon > 0$ there is $a \in A$ such that $d(e(a)(a), g(a)) < \epsilon$. Then every continuous map $f : B \to B$ has approximate fixed points: for every $\epsilon > 0$ there is $b \in B$ such that $d(b, f(b)) < \epsilon$.
Proof. Given any $f$ and $\epsilon$ consider the map $g(a) = f(e(a)(a))$. there is $a \in A$ such that $d(e(a)(a), g(a)) < \epsilon$ and then $b = e(a)(a)$is an $\epsilon$-approximate fixed point of $f$. QED.
One way to use the theorem is via the sup metric (allowing infinite distance):
Corollary: If $e : A \to B^A$ has a dense image in the sup metric on $B^A$ then every endomap on $B$ has approximate fixed points.
Suppose we could apply the previous theorem to the closed ball $D^n$. Then we would know (constructively!) that every endomap on $D^n$ has approximate fixed points. then we just have another easy step, which contains all the classical reasoning needed:
Theorem: Suppose $X$ is compact and $f : X \to X$ has an $\epsilon$-approximate fixed point for every $\epsilon > 0$. Then $f$ has a fixed point.
Proof. By countable choice, for every $n$ there is $x_n \in X$ such that $d(x_n, f(x_n)) < 1/n$. Because $X$ is compact, $x_n$ has a subsequence converging to some $y \in X$. It is now easy to see that $y$ is a fixed point of $f$. QED.
But I do not see how to apply the approximate Lawvere to the closed ball, if that is even possible.
You are correct in observing the flaw in the claims for BFPT to be constructive: There is no algorithm that takes a sequence in the unit hypercube and outputs some accumulation point of it. This task is in fact LESS(1) constructive that BFPT itself. We can be slightly less wasteful, and come up with a sequence converging to some fixed point of a function, but still, computing the limit of a converging sequence is less constructive thatn BFPT.
François has already explained how accepting BFPT compels us to also accept LLPO via IVT. However, IVT is in a sense more constructive than the more general BFPT: Any computable function $f : [0,1] \to [-1,1]$ with $f(0) = -1$ and $f(1) = 1$ has a computable root. However, a computable function $f : [0,1]^2 \to [0,1]^2$ can fail to have any computable fixed points at all.
A framework to compare how constructive certain theorems are is found in Weihrauch reducibility, and Brouwer's Fixed Point theorem is discussed in detail in:
http://arxiv.org/abs/1206.4809
[1] For this, see http://arxiv.org/abs/1101.0792
Best Answer
There is a completely elementary and very elegant proof of the Brower fixed point theorem based on a beautiful combinatorial result called Sperner lemma. For details I recommend Section 2.3, page 72 of the beautiful book
The proof is constructive and it leads to an algorithm for generating a sequence of points converging to a fixed point of the map. Prasolov attributes this approach to
Note This does not really answer your question.